Mathisson–Papapetrou–Dixon equations

In physics, specifically general relativity, the Mathisson–Papapetrou–Dixon equations describe the motion of a massive spinning body moving in a gravitational field. Other equations with similar names and mathematical forms are the Mathisson–Papapetrou equations and Papapetrou–Dixon equations. All three sets of equations describe the same physics.

They are named for M. Mathisson,[1] W. G. Dixon,[2] and A. Papapetrou.[3]

Throughout, this article uses the natural units c = G = 1, and tensor index notation.

Mathisson–Papapetrou–Dixon equationsEdit

The Mathisson–Papapetrou–Dixon (MPD) equations for a mass   spinning body are

 
 

Here   is the proper time along the trajectory,   is the body's four-momentum

 

the vector   is the four-velocity of some reference point   in the body, and the skew-symmetric tensor   is the angular momentum

 

of the body about this point. In the time-slice integrals we are assuming that the body is compact enough that we can use flat coordinates within the body where the energy-momentum tensor   is non-zero.

As they stand, there are only ten equations to determine thirteen quantities. These quantities are the six components of  , the four components of   and the three independent components of  . The equations must therefore be supplemented by three additional constraints which serve to determine which point in the body has velocity  . Mathison and Pirani originally chose to impose the condition   which, although involving four components, contains only three constraints because   is identically zero. This condition, however, does not lead to a unique solution and can give rise to the mysterious "helical motions" .[4] The Tulczyjew–Dixon condition   does lead to a unique solution as it selects the reference point   to be the body's center of mass in the frame in which its momentum is  .

Accepting the Tulczyjew–Dixon condition  , we can manipulate the second of the MPD equations into the form

 

This is a form of Fermi–Walker transport of the spin tensor along the trajectory – but one preserving orthogonality to the momentum vector   rather than to the tangent vector  . Dixon calls this M-transport.

See alsoEdit

ReferencesEdit

NotesEdit

  1. ^ M. Mathisson (1937). "Neue Mechanik materieller Systeme". Acta Physica Polonica. 6. pp. 163–209.
  2. ^ W. G. Dixon (1970). "Dynamics of Extended Bodies in General Relativity. I. Momentum and Angular Momentum". Proc. R. Soc. Lond. A. 314 (1519): 499–527. Bibcode:1970RSPSA.314..499D. doi:10.1098/rspa.1970.0020.
  3. ^ A. Papapetrou (1951). "Spinning Test-Particles in General Relativity. I". Proc. R. Soc. Lond. A. 209 (1097): 248–258. Bibcode:1951RSPSA.209..248P. doi:10.1098/rspa.1951.0200.
  4. ^ L. F. O. Costa; J. Natário; M. Zilhão (2012). "Mathisson's helical motions demystified". AIP Conf. Proc. AIP Conference Proceedings. 1458: 367–370. arXiv:1206.7093. doi:10.1063/1.4734436.

Selected papersEdit